Line searching techniques

ABSTRACT

Techniques of line-searching to find a minimum value of a loss function l projected onto a line are disclosed, wherein the loss function l is continuous and piece-wise linear. One technique includes identifying points for at least one output derived from the loss function l at which the slope of the output changes, determining at least one derivative value for an initial point and at least one change in derivative value for at least some of the points using a processor, determining a cumulative reduction in loss for at least some of the points using at least one of the determined derivative values or at least one of the change in derivative values, and selecting a point corresponding to the minimum value of the loss function l based on the cumulative reduction in loss.

TECHNICAL FIELD

The present invention relates in general to machine learning, multi-label classification, and line searching.

BACKGROUND

Machine learning, a type of artificial intelligence, includes the study of computing devices that are able to recognize patterns in data. One way to enable a computing device to recognize patterns in data (and assign labels to tokens of data) is to train the computing device using a training set of data, commonly called a training corpus. The computing device can then analyze the training set of data to create a classifier model usable to analyze other data sets. In certain cases, the analysis can be performed, in part, using a line search.

A commonly used training set of data for machine learning is the Penn Treebank II corpus. This corpus includes news articles that are annotated with tags (labels) indicating the parts of speech (POS) of the words within the corpus. The labels provide the computing device with “correct answers” for the training set of data that permits the generation of a classifier model during a training process.

SUMMARY

Disclosed herein are embodiments of techniques for line searching.

One aspect of the disclosed embodiments is a technique of line-searching to find a minimum value of a loss function

projected onto a line, wherein the loss function

incorporates a multi-class L₁-regularized hinge loss function that is continuous and piece-wise linear. The technique includes identifying points along the line with respect to an unregularized loss output and a regularization penalty output, the outputs each derived from the loss function

, the points identified based on changes in slope of the outputs at the points, determining at least one derivative value for an initial point of the outputs and at least one change in derivative value for at least some of the points of the outputs using a processor, determining a cumulative reduction in loss for at least some of the points using, for the initial point, the determined derivative values of the outputs at that point, and for points other than the initial point, the change in derivative values at that point, and selecting the point corresponding to the minimum value of the loss function

based on a largest cumulative reduction in loss.

Another aspect of the disclosed embodiments is a technique of line-searching to find a minimum value of a loss function

projected onto a line, wherein the loss function

is continuous and piece-wise linear. The technique includes identifying points for at least one output derived from the loss function

at which the slope of the output changes, determining at least one derivative value of an initial point and at least one change in derivative value of at least some of the points using a processor, determining a cumulative reduction in loss for at least some of the points using at least one of the determined derivative values or at least one of the change in derivative values, and selecting a point corresponding to the minimum value of the loss function

based on the cumulative reduction in loss.

Another aspect of the disclosed embodiments is a technique of line-searching to find an optimum step size value using a loss function with respect to an update vector, wherein the loss function is continuous and piece-wise linear. The technique includes performing for each of at least some examples in a training set steps including identifying a set of step sizes greater than zero along the update vector at which a highest scoring label of one or more examples changes, determining an increment to a derivative of loss corresponding to a step size equal to 0, and determining an increment to a change in the derivative of loss with respect to at least some of the identified step sizes greater than zero. The technique also includes determining a cumulative reduction in loss for at least some of the set of step sizes using at least one of the determined derivatives of loss or at least one of the determined changes in derivatives of loss and selecting the optimum step size value based on the cumulative reduction in loss.

Another aspect of the disclosed embodiments is a computing scheme for line-searching to find a minimum value of a loss function

, wherein the loss function

is continuous and piece-wise linear. The computing scheme includes one or more computing devices each having at least one memory and at least one processor, the computing devices configured to execute instructions stored in at least one memory to: identify points for at least one output derived from the loss function

at which the slope of the output changes, determine at least one derivative value of an initial point and at least one change in derivative value of at least some of the points using a processor, determine a cumulative reduction in loss for at least some of the points using at least one of the determined derivative values or at least one of the changes in derivative values, and select a point having the minimum value of the loss function

based on the cumulative reduction in loss.

These and other embodiments will be described in additional detail hereafter.

BRIEF DESCRIPTION OF THE DRAWINGS

The description herein makes reference to the accompanying drawings wherein like reference numerals refer to like parts throughout the several views, and wherein:

FIG. 1 is a diagram of a computing scheme according to embodiments of the disclosed subject matter;

FIG. 2 is a flowchart of a technique for feature weight training according to embodiments of the disclosed subject matter;

FIG. 3 is a flowchart of a technique for determining updates to feature weights in an update set to reduce loss associated with a classifier model within the technique of FIG. 2 according to embodiments of the disclosed subject matter;

FIG. 4 is a graph illustrative of hinge loss values for individual training examples and for a sum of hinge losses over the training set with respect to a selected feature weight used within the technique of FIG. 3 according to embodiments of the disclosed subject matter;

FIG. 5 is a flowchart of a technique for determining a step size for an update vector to reduce the overall loss of the updates in the update vector within the technique of FIG. 2 according to embodiments of the disclosed subject matter; and

FIG. 6 is a block diagram of a training and classification scheme implemented in the computing scheme of FIG. 1 according to embodiments of the disclosed subject matter.

DETAILED DESCRIPTION

A line search is a technique for finding a minimum value of a function. In the context of machine learning, a line search can be used to minimize the output of a loss function to provide an update to a feature weight to improve the accuracy of a classifier model. In the disclosed subject matter, a line search is provided that is applicable to continuous piece-wise linear functions. The line search is performed in two phases. In the first phase, it captures derivative values at an initial point on a line corresponding to a current classifier model and changes in derivative values at points along a line where the slope of the loss function changes. These values are utilized in phase two to determine a minimum based on cumulative reduction in loss values determined for the points.

FIG. 1 is a diagram of a computing scheme 10 according to embodiments of the disclosed subject matter. An exemplary computing device 12 can be, for example, a computer having an internal configuration of hardware including a processor such as a central processing unit (CPU) 14 and a memory 16. CPU 14 can be a controller for controlling the operations of computing device 12. The CPU 14 is connected to memory 16 by, for example, a memory bus. Memory 16 can include random access memory (RAM) or any other suitable memory device. Memory 16 can store data and program instructions which are used by the CPU 14. Other suitable implementations of computing device 12 are possible.

The computing device 12 can be the only computing device or can be one of a group of computing devices 17 that includes additional computing devices 18. The group of computing devices 17 can be implemented as a computing cluster whereby the computing device 12 and additional computing devices 18 share resources, such as storage memory, and load-balance the processing of requests to the computing device. The group of computing devices 17 can alternatively be a cloud computing service. For example, a cloud computing service can include hundreds or thousands of computing devices configured to provide scalable computing resources. In a cloud computing service, computing tasks can be performed on one or more computing devices or other computing devices included within the cloud computing service.

The above are only exemplary implementations of the group of computing devices 17, and any distributed computing model can be used in their place. As used herein, the term computing device is understood to include any combination or implementation of computing devices, computing device groups, or any other configuration of computing devices.

Other implementations of the computing scheme 10 are possible. For example, one implementation can omit the group of computing devices 17 and additional computing devices 18 and include only a single computing device 12.

In one example of machine learning, a classification problem can be defined using a label universe

and a feature universe

. A classification problem is solved using a classifier. The label universe

contains all labels that can be assigned by the classifier. Each token (piece of data) processed by the classifier is assigned at least one label by the classifier based on at least one feature associated with that piece of data. In the implementation described herein, labels include parts of speech (POS). For example, POS can include words (tokens) that are nouns, verbs, and adjectives. However, in alternative implementations, any universe of labels

can be utilized (for example, labels indicating a category of a news article). Tokens can be, for example, one word. Alternatively, tokens can be created based on any portion of a dataset.

Feature universe

contains all features that can be used by the classifier. Features are intrinsic to the data, meaning that they are objectively determinable by the classifier for each token extracted from a dataset. For example, some features in feature universe

can include: Contains a digit, Contains a hyphen, Contains an upper-case character, Lower-cased frequency class, Original-case frequency class, Lower-cased prefixes up to 4 characters, Lower-cased suffixes up to 4 characters, Lower-cased token, and Original-case token. The above features can be evaluated for each of the previous, current, and next tokens. Additional features can include whether the current token follows or precedes a sentence boundary. Each token has an associated feature set f that contains the features from feature universe

applicable to that token, such that f

.

The features described above are indicator features, meaning that they are binary in nature. For the above features, an example/token either has the feature (in which case, the value of the feature can be 1) or it does not (in which case, the value of the feature can be 0). However, features can include features having other integer or floating point values. For example, in a classifier intended to classify new stories by topic, a token might be an entire news story, and an integer-valued feature might be the number of times a particular word occurs in the new story. While the exemplary implementations are described with respect to the use of binary features, other integer and floating point valued features can also be utilized with the processes described herein.

A classifier model w is defined as a vector wε

. Classifier model w includes feature weight coordinates w_((f,l)) for at least some feature f and at least some label l. The value of each feature weight coordinate is, for example, a floating point number indicative of whether the existence of feature f provides evidence that label l is the best label for a token. The classifier model w is used to predict a label l of a token by summing together, for each label, the feature weights corresponding to the set f of features associated with the token and selecting the label l having the maximum summed value as defined in formula 1:

$\begin{matrix} {{l\left( {f,w} \right)} = {\underset{l \in}{argmax}{\sum\limits_{f \in f}w_{({f,l})}}}} & (1) \end{matrix}$

FIG. 2 is a flowchart of a technique 50 for feature weight training according to embodiments of the disclosed subject matter. Feature weight training is the process of determining or generating classifier model w. The technique 50 can be referred to as Simultaneous MERT (SiMERT). SiMERT is in part related to Minimum Error Rate Training (MERT), a process for optimizing low-dimensional classifier models. SiMERT provides, in one use-case, the ability to scale for use in a high-dimensional classifier including, for example, hundreds of thousands of features. Technique 50 can be more efficient than conventional techniques, particularly when a large number of feature weights are considered. Conventional techniques can include, for example, sequential co-ordinate descent or Powell's Method, where only one feature weight update is computed per iteration. Also, technique 50 can be utilized with non-convex loss functions, unlike some conventional techniques that can only be utilized with convex loss functions (i.e. those used to process large numbers of feature weights).

In general, feature weight training includes the process of taking a training set T that includes annotations of “correct” labels for each token and using the training set T to generate the classifier model w. A goal of feature weight training is to produce a classifier model w that accurately identifies the labels of tokens in data other than the training set (i.e. without the use of annotations). For testing purposes, a set of common test data can be used to quantitatively measure the accuracy of classifier model w (i.e. identify a percentage of labels accurately identified).

Starting at stage 52, technique 50 includes initializing classifier model (vector) w and calculating initial scores for vector w using training set T. Vector w can be initialized to include all zero feature weights. Correspondingly, the initial scores will all be zero, since all of the weights are zero. However, in an alternative implementation, vector w can be initialized to at least some non-zero values. At stage 54, an initial update set U⊂ w of feature weights is selected. The initial update set U of feature weights indicates which feature weights will be updated with new values in the first iteration of the training process.

Limiting the update set U to a subset of w can provide the benefit of producing a sparser model w. In other words, sparse means that a significant number of feature weights w_((f,l)) are zero. If the model w is initialized to zero for all feature weights, then the number of feature weights that become non-zero is proportional to the number of feature weights that can be updated in the iterative updates to w, especially in the first few updates.

One technique for selecting the initial update set U includes using a log-likelihood-ratio (LLR) association metric. In this technique, an LLR score is calculated for each feature-value pair (f,l) found in the training set T. The feature-value pairs are ordered by their scores, and a certain number are selected to include their associated feature weights in update set U in order from the highest score. The certain number can be pre-determined or be otherwise chosen or determined by technique 50. The feature weights included in update set U can be restricted to those that have a negative hinge-loss derivative. An alternative technique for selecting the initial update set includes using a hinge-loss derivative instead of the LLR.

Once the update set U is selected, an update u is determined independently for each feature-value pair (f,l) included in U at stage 56. The update u is determined based on a loss function

. Loss function

can be any function of a set of feature weights such that a decrease in the output value of the function generally indicates an increase in the quality of the predictions made by a model using those feature weights. Loss functions can include, for example, L₁-regularized hinge loss, L₂-regularized hinge loss, or a simple error count with respect to a training set. Update u can be calculated as a value u′ that minimizes the output of loss function

using formula 2:

$\begin{matrix} {{u_{({f,l})} = {\underset{u^{\prime}}{\arg\;\min}\left( {w + {u^{\prime}b_{({f,l})}}} \right)}};{wherein}} & (2) \end{matrix}$ wherein b is a vector of the same dimension as w, wherein b has a value of 1 at coordinate (f,l) and a value of 0 at all other coordinates.

Formula 2 can be computed using, for example, a line-search technique. Once an update u is determined for each feature-value pair (f,l) included in U, the updates u are combined into an update vector u at stage 58. The combination can be performed using formula 3:

$\begin{matrix} {u = {\sum\limits_{f,l}{u_{({f,l})}b_{({f,l})}}}} & (3) \end{matrix}$

After update vector u is created from updates u, a step size α is determined for weighting the update vector u at stage 60. The combination of updates u will generally result in an “overshooting” of the targeted minimized loss (resulting in a higher loss than targeted). This is because feature weights are interdependent (i.e. the selection of a label is the result of summing together multiple feature weights) and the updates u are each selected independently from each other. As such, step size α is selected to weight the updates u as a group in an effort to prevent overshooting the targeted minimized loss. Step size α is generally a value between 0 and 1. The step size α can often be between the values of 0.1 and 0.25, although α is typically 0.5 on the first iteration of the training process. The determination of step size α as a value α′ that minimizes loss function

can be expressed mathematically as shown in formula 4:

$\begin{matrix} {\alpha = {\underset{\alpha^{\prime} > 0}{{argmin}\;}\left( {w + {\alpha^{\prime}u}} \right)}} & (4) \end{matrix}$

The step size α can be determined using line-search techniques. Once the step size α is determined, at stage 62, feature weights in vector w are updated using the step size α and update vector u. The updated vector w′ replaces vector w for subsequent stages in technique 50. Updated vector w′ can be determined using formula 5: w′=w+αu  (5)

At stage 64, technique 50 determines whether to stop the training process or to continue iterating to further update vector w. Various techniques can be used to identify when the training process should be terminated. One technique includes testing the classifier model w produced by each iteration of the training process. The testing can include the use of a held-out data set. The held-out data set includes annotated tags (“correct” labels). The classifier model w is used to determine at least one label for each token in the held-out data set, and the determined labels are compared to the “correct” labels to determine the accuracy of classifier model w.

The technique of testing can be extended by storing a certain number of test results from past iterations in order to determine when the accuracy of classifier model w has been maximized. For example, the accuracy results from the past three or five iterations can be stored. However, any number of past accuracy results can be stored in alternative implementations. If, for an identified number of iterations, the accuracy does not improve, then the most accurate model found will be used. For example, if the identified number is three, once three iterations are completed without improvement, the most accurate model recently found (in this case, from four iterations ago) will be selected, and the technique 50 for training will complete.

Alternatively, other techniques for stopping the training technique 50 can also be used. For example, instead of using held-out data, the loss reduction on the training set between iterations can be measured and monitored. If the loss reduction falls below a threshold for a certain number of iterations, such as 1-5 iterations, then the stoppage of the training process can be triggered.

If the training process is not stopped at the current iteration, control passes to stage 66, where updated scores are determined for the current model w (same as w′ from stage 62) using the training set T. The determined scores can include calculating the absolute value of the derivative of a loss function

.

At stage 68, an updated update set U of feature weights is determined based on the updated scores. In an exemplary implementation, updated update set U is determined for each iteration of the training technique 50. The inclusion of a feature weight in update set U in a previous iteration does not guarantee its inclusion in update set U in later iterations. This contrasts with previous machine learning techniques, which simply add to previously selected update sets. Also, the number of feature weights included in update set U is typically increased for each iteration of technique 50. For example, an initial update set U can contain 1,000 feature weights, while subsequent iterations can each include an additional 1,000 feature weights.

However, in some implementations and situations, subsequent iterations will not include a greater number of feature weights in update set U. For example, in some iterations, there may be no additional feature weights that qualify for selection into update set U. In one exemplary situation, remaining feature weights may have a score of zero or other score beneath a selection threshold.

Starting with the highest score, feature weights are selected into update set U until the number of feature weights desired for update set U is reached, or no additional feature weights having a score greater than the selection threshold are available. In an exemplary implementation, the score is the absolute value of the derivative of a loss function and the selection threshold is 0. Once the updated update set U is determined, control returns to stage 56 for the next iteration of technique 50 using updated update set U and updated model w.

The steps of technique 50 represent only one implementation of technique 50, and various alternative implementations of technique 50 are available, including those that add, delete, and modify steps of technique 50. Some of the steps of technique 50 have been described without regard to exact technical implementation and may require additional steps to complete. For example, certain steps may utilize technique 150 and technique 180 as described later. However, alternative processes may be used instead of technique 150 and technique 180. The formulas herein describe a model for achieving the technique 50. However, implementations of technique 50 may not actually compute each of the formulas shown or may reduce in some way the computation of said formulas for convenience, computational complexity reduction, or various other reasons.

FIG. 3 is a flowchart of a technique 150 for determining updates to feature weights in an update set to reduce loss associated with a classifier model within the technique 50 of FIG. 2 according to embodiments of the disclosed subject matter. Specifically, technique 150 can be utilized to determine independent updates within stage 56. Within technique 150, there are two phases, phase I 154 and phase II 156. In phase I 154, values are collected for the line searches of phase II 156 using the scores for each example in the training set. In phase II 156, the values collected in phase I 154 for each feature weight are used in line-searches to determine update u.

Technique 150 will be described using an L₁-regularized loss function

. However, technique 150 is not limited to use of this particular loss function

and may utilize alternative loss functions

. Loss function

will typically be continuous and piece-wise linear. The L₁-regularized loss function includes the use of multiclass hinge loss. Hinge loss is the amount by which a score of a correct label fails to exceed, by at least a margin of 1, the score of any other label. It is calculated by taking the larger of zero or one plus the difference between the largest sum of the feature weights of an incorrect label and the sum of the feature weights of the correct label. Hinge loss hl(w; (f,l_(h))) of an example token from a training set T can be defined mathematically as shown in formula 6:

$\begin{matrix} {{{{hl}\left( {w;\left( {f,l_{h}} \right)} \right)} = {\max\left( {0,{1 + {\max\limits_{l_{h}^{\prime} \neq l_{h}}{\sum\limits_{f \in f}w_{({f,l_{h}^{\prime}})}}} - {\sum\limits_{f \in f}w_{({f,l_{h}})}}}} \right)}};} & (6) \end{matrix}$ wherein l_(h) is the correct label of the example token; and l_(h)′ ranges over the incorrect labels (i.e all labels in

except for l_(h)).

The L₁-regularized loss function

can be defined in terms of the hinge loss. The L₁-regularized loss function

includes a first term that is the unregularized loss value and a second term that is the regularization value. The unregularized loss value is determined using a sum of the hinge losses for each token (or subset of tokens) in training set T. The regularization value prevents the model w from overfitting the training set T. Overfitting occurs when the loss on the examples included in T is optimized to the point that there is a lower classification accuracy on other examples not included in T. The L₁-regularized loss function

is shown in formula 7:

$\begin{matrix} {{{(w)} = {{\sum\limits_{{({f,l_{h}})} \in T}{{hl}\left( {w;\left( {f,l_{h}} \right)} \right)}} + {C{\sum\limits_{f \in}{\sum\limits_{l_{r} \in}{w_{({f,l_{r}})}}}}}}};} & (7) \end{matrix}$ wherein C is a regularization constant that controls a balance between regularization and unregularized loss reduction

Formula 7 can be used as a basis to identify an unregularized loss for an update u to a particular feature weight and a regularization value for the update u to the feature weight. The unregularized loss is determined using a sum of the hinge losses for each token (or subset of tokens) in training set T with respect to u. Since the hinge loss on a particular example is piecewise linear, the sum of hinge loss over a set of examples will also be piecewise linear. A function l_((f,l) _(u) ₎(u) for determining an unregularized loss for the update u can be projected from the L₁-regularized loss function

as shown in formula 8 and a function r_((f,l) _(u) ₎(u) for determining a regularization value for the update u can be projected from the L₁-regularized loss function

as shown in formula 9:

$\begin{matrix} {{{l_{({f,l_{u}})}(u)} = {\sum\limits_{{({f,l_{u}^{\prime}})} \in T}{{hl}\left( {{w + {ub}_{({f,l_{u}})}};\left( {f,l_{u}^{\prime}} \right)} \right)}}};{wherein}} & (8) \end{matrix}$ wherein l_(u) is the label of the feature weight to be updated by u; l_(u)′ is the correct label for the example (f,l_(u)′); b is a vector having a value of 1 at coordinate (f,l_(u)) and a value of 0 at all other coordinates. r _((f,l) _(u) ₎(u)=C|w _((f,l) _(u) ₎ +u|  (9)

FIG. 4 is a graph 200 illustrative of hinge loss values for individual training examples and for a sum of hinge losses over the training set with respect to a selected feature weight used within the technique 150 of FIG. 3 according to embodiments of the disclosed subject matter. Graph 200 includes a sum-of-losses plot 202 and individual-example hinge loss plots 204 a-f. Update plot 202 corresponds to the output of formula 8. Hinge losses 204 a-f correspond to hinge loss values within the summation of formula 8 for various tokens within the training set T (i.e. various examples of (f,l_(u)′)εT). The graph 200 shows that some hinge losses can have a positive slope while others can have a negative slope. The slope of a hinge loss graph is positive if the selected feature weight is associated with an incorrect label for an example/token. Correspondingly, the slope is negative if the selected feature weight is associated with a correct label for an example/token. In other words, the direction in which the hinge loss increases for a particular feature weight associated with a particular label is correlated with whether that label is the correct label for the example/token in training set T for which the hinge loss is measured.

When summed together, the hinge loss plots 204 a-f become plot 202, which, in this example, is continuous piecewise linear and mathematically convex. Plot 202 includes vertices 206 a-e that each correspond with at least one vertex present in the hinge loss plots 204 a-f. Vertices 206 a-e are the data points on which the steps of technique 150 (and similarly, technique 180) operates.

Referring back to FIG. 3, in phase I 154, values are collected for the line searches using the scores for each example in the training set. Referring to phase I 154 conceptually, the following sets of values are collected for each feature weight w_((f,l) _(u) ₎ in the update set. The values dl_((f,l) _(u) ₎(0) as the derivative

$\frac{\mathbb{d}{l_{({f,l_{u}})}(u)}}{\mathbb{d}u}$ of l_((f,l) _(u) ₎(u) at an initial point where u=0 and dr_((f,l) _(u) ₎(0) as the derivative

$\frac{\mathbb{d}{r_{({f,l_{u}})}(u)}}{\mathbb{d}u}$ of r_((f,l) _(u) ₎(u) at an initial point where u=0 can be collected. There need not be a change in the derivative at the initial point. However, values relating to the regularization value r can be collected during stage 162 or stage 164 of phase II 156 to improve efficiency. dl_((f,l) _(u) ₎(0) includes a forward dl_((f,l) _(u) ₎ ⁺(0) derivative measurement, used when searching values of u>0 (i.e. a forward line search), and a backwards dl_((f,l) _(u) ₎ ⁻(0) derivative measurement, used when searching values of u<0 (i.e. a backward line search). Similarly, dr_((f,l) _(u) ₎(0) includes a forward dr_((f,l) _(u) ₎ ⁺(0) derivative measurement and a backwards dr_((f,l) _(u) ₎ ⁻(0) derivative measurement. Each is used for one direction of line searching in phase II 156. At an inflection point of a continuous piecewise linear function, the forward and backward derivative measurements for the function will have different values. When the term “derivative” is used with respect to a continuous piecewise linear function at such a point, it refers to either the forward or backward derivative measurement for the function at that point depending on whether the reference occurs within the context of a forward search or a backward search.

For each value u, at which there is a vertex (inflection) point in the loss function, Δdl_((f,l) _(u) ₎(u_(i)) is determined and collected as the change in

$\frac{\mathbb{d}{l_{({f,l_{u}})}(u)}}{\mathbb{d}u}$ at u_(i). Further, Δdr_((f,l) _(u) ₎(u_(i)) is determined and collected as the change in

$\frac{\mathbb{d}{r_{({f,l_{u}})}(u)}}{\mathbb{d}u}$ at an additional point u_(i)=−w_((f,l) _(u) ₎.

In one implementation, at stage 157, an example is selected from the training set T. At stage 158, values are collected for the feature weights present in the selected example. At stage 159, if there are additional training set examples available, control returns to stage 157 to select the next available example. Otherwise, the collection of values is complete, and control passes to the first stage of phase II 156 (stage 160).

While the values collected for a particular feature weight can be determined using a number of techniques, the following is a technique that can be used by some implementations, such as within stages 157-159 above. The collected values can be determined based on training examples/tokens in training set T. The updates made for each training example/token are based on difference Δ_(s). Difference Δ_(s) is the difference between the score for the correct label of the example/token and the highest score of the incorrect labels.

For a given training example/token and possible label, we define a relevant feature weight to be a feature weight contained in the current update set that corresponds to the label combined with a feature that is exemplified by the example/token. In stage 158, if Δ_(s)=1, then (1) dl_((f,l) _(u) ₎ ⁻(0) is decremented by 1 for the relevant feature weights corresponding to the correct label for the selected example/token; and (2) dl_((f,l) _(u) ₎ ⁺(0) is incremented by 1 for relevant feature weights corresponding to the highest-scoring incorrect labels for the selected example/token.

Otherwise, if Δ_(s)<1, then (1) both dl_((f,l) _(u) ₎ ⁺(0) and dl_((f,l) _(u) ₎ ⁻(0) are decremented by 1 for the relevant feature weights corresponding to the correct label for the selected example/token; (2) dl_((f,l) _(u) ₎ ⁺(0) is incremented by 1 for the relevant feature weights corresponding to the highest-scoring incorrect labels for the selected example/token; and (3) dl_((f,l) _(u) ₎ ⁻(0) is incremented by 1 for the relevant feature weights corresponding to the highest-scoring incorrect label for the selected example/token, if no other incorrect label has the same score.

Increments to instances of Δdl_((f,l) _(u) ₎(u_(i)) can be determined by a simple criterion. In stage 158, for feature weights w_((f,l) _(u) ₎ in the update set, Δdl_((f,l) _(u) ₎(u_(i)) increases by 1 if an update of u_(i) to feature weight w_((f,l) _(u) ₎ changes the minimum difference on the selected training example between the score for the correct label and the score for any incorrect label, such that minimum difference becomes exactly 1. Since each update affects only a single label, the relevant updates can be found simply by looking at differences in scores for pairs of labels.

Accumulating the updates for all instances of Δdl_((f,l) _(u) ₎(u_(i)) can be the most memory intensive aspect of SiMERT. To reduce memory requirements, the updates u_(i) can be quantized at values of 1, −1, and multiplicative increments and decrements of 1.1. That is, every update is represented by an element of the sequence ( . . . , 1.1⁻², 1.1⁻¹, 1, 1.1¹, 1.1², . . . ) or the sequence ( . . . , −(1.1⁻²), −(1.1⁻¹), −1, −(1.1¹), −(1.1²), . . . ). Due to this quantization, the feature weights can be limited to no more than a few hundred distinct quantized update values on any iteration, including feature weights with more than 100,000 training data occurrences.

While

$\frac{\mathbb{d}{r_{({f,l_{u}})}(u)}}{\mathbb{d}u}$ at u=0 for a particular feature weight can be determined using a number of techniques, the following is a technique that can be used by some implementations, such as within stages 162 or 164 below or 157-159 above. As mentioned previously, values related to regularization value r can be collected during phase II 156 in some implementations to improve efficiency.

$\frac{\mathbb{d}{r_{({f,l_{u}})}(u)}}{\mathbb{d}u}$ at u=0 includes a forward dr_((f,l) _(u) ₎ ⁺(0) and a backwards dr_((f,l) _(u) ₎ ⁻(0) derivative measurement. For a feature weight in the update set having a current value w₀, the derivative values can be determined by (1) if w₀>0, then dr_((f,l) _(u) ₎ ⁺(0) and dr_((f,l) _(u) ₎ ⁻(0) are each set to C (the regularization constant); (2) if w₀=0, then dr_((f,l) _(u) ₎ ⁺(0) is set to C and dr_((f,l) _(u) ₎ ⁻(0) is set to −C; and (3) if w₀<0, then dr_((f,l) _(u) ₎ ⁺(0) and dr_((f,l) _(u) ₎ ⁻(0) are each set to −C. Δdr_((f,l) _(u) ₎(u) is defined only for u=−w₀, such that Δdr_((f,l) _(u) ₎(−w₀) is set to 2C.

The values collected in phase I 154 are used in the line searches of phase II 156. For phase II 156, at stage 160, a feature weight is selected for which an update u will be determined. For each selected feature weight, two line searches are performed, one through the positive values of u_(i) collected for the feature weight and one through the negative values of u_(i) collected for the feature weight. At stage 162, phase II variables are initialized with respect to each line search for the selected feature weight. The variables include initial cumulative reduction in loss Δ

_(,0) and variables dl and dr, which track

$\frac{\mathbb{d}{l_{({f,l_{u}})}(u)}}{\mathbb{d}u}$ and

$\frac{\mathbb{d}{r_{({f,l_{u}})}(u)}}{\mathbb{d}u}$ for each value of u_(i).

For both search directions, initial cumulative reduction in loss change Δ

_(,0) is set to zero. For the positive (forward) search direction, dl is set to dl_((f,l) _(u) ₎ ⁺(0) from phase I 154, and dr is set to dr_((f,l) _(u) ₎ ⁺(0) from phase I 154. For the negative (backward) search direction, dl is set to dl_((f,l) _(u) ₎ ⁻(0) from phase I 154, and dr is set to dr_((f,l) _(u) ₎ ⁻(0) from phase I 154. Next, at stage 164, technique 150 iterates through the collected data points u_(i), in order, for each of the forwards and backwards line searches to determine the cumulative reduction for each point. Points u_(i) that have a value of Δdl(u_(i)) and Δdr(u_(i)) each equal to zero can be bypassed in the iteration. The cumulative reduction in loss for each data point u_(i) (Δ

_(,i)) is set using formula 10: Δ

_(,i)=Δ

_(,i-1)−(dl+dr)(u _(i) −u _(i-1))  (10)

In addition, dl and dr can also be updated during the iterations of stage 164. If, at a given point u_(i),

$\frac{\mathbb{d}{l_{({f,l_{u}})}(u)}}{\mathbb{d}u}$ changes, dl is updated by adding Δdl_((f,l) _(u) ₎(u_(i)) (collected from phase I 154) to dl. Also, if, at a given point u_(i),

$\frac{\mathbb{d}{r_{({f,l_{u}})}(u)}}{\mathbb{d}u}$ changes, dr is updated by adding Δdr_((f,l) _(u) ₎(u_(i)) (collected from phase I 154) to dr. The updated values of dl and dr will be used for the next iteration of stage 164. If l_((f,l) _(u) ₎(u) and r_((f,l) _(u) ₎(u) are known to be convex functions, then phase II 156 can be configured to terminate early when Δ

_(,i) is not greater than Δ

_(,i-1).

Lastly, at stage 166, a value u_(i) is selected and returned based on a value of i that corresponds to a largest value of cumulative loss reduction Δ

_(,i). The largest value of cumulative loss reduction Δ

_(,i) can be a maximum cumulative loss reduction or an approximation thereof. Correspondingly, the value u_(i) can correspond to a minimum value (or an approximation thereof) of the loss function

for any possible value of u_(i). The value u_(i) is the value u_((f,l)) as shown in formula 2. Finally, at stage 168, technique 150 determines whether there are additional feature weights to be processed. If so, control returns to stage 160. Otherwise, technique 150 ends.

The above describes only one implementation of a technique 150 for determining various updates u for feature weights of model w. Various alternative and modified implementations can also be utilized. For example, certain steps of the process may be performed taking into account all selected feature weights at a time, such as when measurements and calculations are performed during the line searches. The formulas above represent a technique for identifying an update for multiple feature weights within a single iterative process. It is anticipated that variations and modifications of the above formulas and techniques will be made to accommodate various implementations. Also, other types of line searches can be used for loss functions that are not piecewise continuous linear. For example, a MERT line search can be used if the loss function is an error count.

FIG. 5 is a flowchart of a technique 180 for determining a step size α for an update vector u to reduce the overall loss of the updates in the update vector within the technique 50 of FIG. 2 according to embodiments of the disclosed subject matter. Similar to formulas 8-9, a function l(α) for determining an unregularized loss for a step size α can be projected from the L₁-regularized loss function

as shown in formula 11 and a function r(a) for determining a regularization value for a step size α can be projected from the L₁-regularized loss function

as shown in formula 12:

$\begin{matrix} {{l(\alpha)} = {\sum\limits_{{({f,l_{r}})} \in T}{{hl}\left( {{w + {\alpha\; u}};\left( {f,l_{r}} \right)} \right)}}} & (11) \\ {{r(\alpha)} = {C{\sum\limits_{f \in {l_{r}} \in}{{w_{({f,l_{r}})} + {\alpha\; u_{({f,l_{r}})}}}}}}} & (12) \end{matrix}$

Similar to technique 150, technique 180 includes two phases, phase I 182 and phase II 184. During phase I 182, values are collected, including dl and dr, and for various values of α_(i), also Δdl(α_(i)) and Δdr(α_(i)). Phase II 184 includes a line search to find a sufficiently accurate value for step size α. With respect to phase I 182, starting at stage 186 an example is selected from the training set. At stage 187, unregularized loss values associated with the selected example are collected. Then, at stage 188, if there are additional examples available for selection from training set T, control returns to stage 186 to select the next example. Otherwise, regularization values are collected. First, a feature weight is selected from the update set at stage 189. Regularization values are collected with respect to the feature weight at stage 190. Then at stage 191, if there are additional feature weights available for selection, control returns to stage 189 to select the next feature weight. Otherwise, collection of values is complete and control passes to phase II 184 by way of stage 192.

The stages of phase I 182 can be implemented in various different ways, and one potential implementation is disclosed here. Collection and computation of the above values is simplified by subtracting 1 (the minimum desired margin) from the score of the correct label for each training example. This allows for the hinge loss for each example to be treated as merely the difference between the highest score of any label and the score for the correct label. For an example having (f,l_(s)) (wherein l_(s) is the correct label), the score s(l_(s)′) of any label l_(s)′ with respect to the example can be defined using formula 13:

$\begin{matrix} {{{s\left( l_{s}^{\prime} \right)} = {\left( {\sum\limits_{f \in f}w_{({f,l_{s}^{\prime}})}} \right) - {\delta\left( {l_{s}^{\prime},l_{s}} \right)}}};{wherein}} & (13) \end{matrix}$ wherein δ(l_(s)′,l_(s)) is equal to 1 if l_(s)′=l_(s) and 0 otherwise.

Given formula 13, hinge loss can be redefined as shown in formula 14 and, assuming the use of formula 14, the derivatives of the score and the hinge loss can be defined as shown in formulas 15-16:

$\begin{matrix} {{{hl}\left( {w;\left( {f,l_{s}} \right)} \right)} = {{\max\limits_{l_{s}^{\prime}}{s\left( l_{s}^{\prime} \right)}} - {s\left( l_{s} \right)}}} & (14) \\ {\frac{\mathbb{d}{s_{l_{s}^{\prime}}(\alpha)}}{\mathbb{d}\alpha} = {\sum\limits_{f \in f}u_{({f,l_{s}^{\prime}})}}} & (15) \\ {{{\frac{\mathbb{d}{l(\alpha)}}{\mathbb{d}\alpha}\left( \alpha_{i} \right)} = {\frac{\mathbb{d}{s_{l_{s}^{\prime}}(\alpha)}}{\mathbb{d}\alpha} - \frac{\mathbb{d}{s_{l_{s}}(\alpha)}}{\mathbb{d}\alpha}}};{wherein}} & (16) \end{matrix}$ wherein l_(s)′ is the highest scoring label at α_(i) and l_(s) is the correct label.

Based on the above defined formulas 13-16, values of dl(0) (the derivative at an initial point where α=0) and Δdl(α_(i)) can be determined. First, for each training example, the values of α>0 at which the highest scoring label changes are determined using, for example, a MERT line search based on the scores s(l_(s)′). These values are arranged in order to define a sequence α₁, . . . , α_(k). Also defined is a sequence of highest scoring labels l₀, . . . , l_(k), such that α_(i) is the point at which the highest scoring label changes from l_(i-1) to l_(i). For each training example, dl(0) is incremented by

$\frac{\mathbb{d}{s_{l_{0}}(\alpha)}}{\mathbb{d}\alpha} - \frac{\mathbb{d}{s_{l_{s}}(\alpha)}}{\mathbb{d}\alpha}$ and Δdl(α_(i)) is incremented by

$\frac{\mathbb{d}{s_{l_{i}}(\alpha)}}{\mathbb{d}\alpha} - \frac{\mathbb{d}{s_{l_{i - 1}}(\alpha)}}{\mathbb{d}\alpha}$ for each value of i from 1 to k.

To determine the values of dr(0) and Δdr(α_(i)), for each feature weight in the update set having a value w₀ and a determined update u_((f,l)), the following steps are performed. First, if u_((f,l))>0 and w₀≧0, dr(0) is incremented by Cu_((f,l)). Second, if u_((f,l))>0 and w₀<0, dr(0) is decremented by Cu_((f,l)) and Δdr(−w₀/u_((f,l))) is incremented by 2Cu_((f,l)). Third, if u_((f,l))<0 and w₀>0, dr(0) is decremented by −Cu_(l)) and Δdr(−w₀/u_((f,l))) is incremented by −2Cu_((f,l)). Fourth, if u_((f,l))<0 and w₀≦0, dr(0) is incremented by −Cu_((f,l)).

At stage 192, variables for phase II 184 are initialized. Such variables can include initial cumulative loss reduction Δ

_(,0) and intermediate variables dl and dr, which track the derivatives of unregularized loss and regularization value, respectively, with respect to a for each value of α_(i).

Initial cumulative loss reduction Δ

_(,0) is set to zero, dl is set to dl(0) from phase I 182, and dr is set to dr(0) from phase I 182. Next, at stage 193, technique 180 iterates through the collected data points α_(i), in order, for the line search to determine the cumulative reduction for each point. Points α_(i) that have values of Δdl(α_(i)) and Δdr(α_(i)) each equal to zero can be bypassed in the iteration. The cumulative reduction for each data point α_(i) (Δ

_(,i)) is set using formula 17: Δ

_(,i)=Δ

_(,i-1)−(dl+dr)(α_(i)−α_(i-1))  (17)

In addition, dl and dr can also be updated during the iterations of stage 193. If, at a given point α_(i),

$\frac{\mathbb{d}{l(\alpha)}}{\mathbb{d}\alpha}$ changes, dl is updated by adding Δdl(α_(i)) (collected from phase I 182) to dl. Also, if, at a given point α_(i),

$\frac{\mathbb{d}{r(\alpha)}}{\mathbb{d}\alpha}$ changes, dr is updated by adding Δdr(α_(i)) (collected from phase I 182) to dr. The updated values of dl and dr will be used for the next iteration of stage 193. If l_((f,l) _(u) ₎(α) and r_((f,l) _(u) ₎(α) are known to be convex functions, then phase II 184 can be configured to terminate early when Δ

_(,i) is not greater than Δ

_(,i-1).

Lastly, at stage 194, an optimum step size value α_(i) is selected and returned based on a value of i that corresponds to a largest value of cumulative loss reduction Δ

_(,i). The largest value of cumulative loss reduction Δ

_(,i) can be a maximum cumulative loss reduction over possible step sizes or an approximation thereof. Correspondingly, the value α_(i) can correspond to a minimum value (or an approximation thereof) of the loss function

with respect to an update vector u for any value of α_(i)≧0. The value α_(i) is the value α as shown in formulas 4 and 5.

FIG. 6 is a block diagram of a training and classification scheme 100 implemented in the computing scheme 10 of FIG. 1 according to embodiments of the disclosed subject matter.

The scheme 100 includes a trainer 102 that is configured to accept input 104. Input 104 includes a training set T, feature universe

, label universe

and a loss function

. Input 104 is used by trainer 102 to generate a classification model w 106. The generation can be performed, for example, using technique 50. The generated classification model w is output to classifier 108.

Classifier 108 is configured to accept input 110 that includes a feature set f_(n) corresponding to a token n of a data set to be classified. Classifier 108 outputs l_(n), a label for token n selected by classifier 108 based on the feature set f_(n) and classification model w. The selection by classifier 108 can be performed, for example, by use of formula 1.

The trainer 102 and classifier 108 illustrate one possible implementation of scheme 100. Other implementations are available, including those that add, modify, and delete the inputs and outputs to and from trainer 102 and classifier 108.

The embodiments of computing device 12 and/or computing devices 17 (and the algorithms, techniques, instructions etc. stored thereon and/or executed thereby) can be realized in hardware including, for example, computers, IP cores, application-specific integrated circuits (ASICs), programmable logic arrays, optical processors, programmable logic controllers, microcode, firmware, microcontrollers, computing devices, microprocessors, digital signal processors or any other suitable circuit. As used herein, the term “processor” should be understood as encompassing any the foregoing, either singly or in combination. The terms “signal” and “data” are used interchangeably. Further, portions of computing device 12 and/or computing devices 17 do not necessarily have to be implemented in the same manner.

Further, in one example, computing device 12 and/or computing devices 17 can be implemented using a general purpose computer/processor with a computer program that, when executed, carries out any of the respective techniques, algorithms and/or instructions described herein. In addition or alternatively, for example, a special purpose computer/processor can be utilized which can contain specialized hardware for carrying out any of the techniques, algorithms, or instructions described herein.

Implementations or portions of implementations of the above disclosures can take the form of a computer program product accessible from, for example, a computer-usable or computer-readable medium. A computer-usable or computer-readable medium can be any device that can, for example, tangibly contain, store, communicate, or transport the program for use by or in connection with any processor. The medium can be, for example, an electronic, magnetic, optical, electromagnetic, or a semiconductor device. Other suitable mediums are also available.

The exemplary approaches herein have been described in order to allow easy understanding of the disclosed subject matter and do not limit the disclosed subject matter. On the contrary, the disclosed subject matter is intended to cover various modifications and equivalent arrangements included within the scope of the appended claims, which scope is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structure as is permitted under the law. 

What is claimed is:
 1. A method of line-searching to find a minimum value of a loss function

projected onto a line, wherein the loss function

incorporates a multi-class L₁-regularized hinge loss function that is continuous and piece-wise linear, the method comprising the steps of: identifying points along the line with respect to an unregularized loss output and a regularization penalty output, the outputs each derived from the loss function

, the points identified based on changes in slope of the outputs at the points; determining at least one derivative value for an initial point of the outputs and at least one change in derivative value for at least some of the identified points of the outputs using a processor; determining a cumulative reduction in loss for at least some of the points of the outputs using, for the initial point, the determined derivative values of the outputs at the initial point, and for points other than the initial point, the change in derivative values at the identified point other than the initial point; and selecting the point corresponding to the minimum value of the loss function

based on a largest cumulative reduction in loss.
 2. A method of line-searching to find a minimum value of a loss function

projected onto a line, wherein the loss function

is continuous and piece-wise linear, the method comprising: (A) identifying points for at least one output derived from the loss function

at which the slope of the output changes; (B) determining at least one derivative value of an initial point and at least one change in derivative value of at least some of the points using a processor; (C) determining a cumulative reduction in loss for at least some of the points using at least one of the determined derivative values or at least one of the change in derivative values; and (D) selecting a point corresponding to the minimum value of the loss function

based on the cumulative reduction in loss.
 3. The method of claim 2, wherein the loss function

is regularized.
 4. The method of claim 3, wherein the loss function

is a multi-class L₁-regularized hinge loss that is derivable from the formula: ${{(w)} = {{\sum\limits_{{({f,l_{h}})} \in T}{{hl}\left( {w;\left( {f,l_{h}} \right)} \right)}} + {C{\sum\limits_{f \in}{\sum\limits_{l_{r} \in}{w_{({f,l_{r}})}}}}}}};$ wherein

is a universe of features; f is a single feature; l_(r) is a single label; f is a set of features associated with an example;

is a universe of labels; T is a training set including examples having a set of features f and a correct label l_(h); (f,l_(h)) is a feature set-label pair associated with an example; hl(w;(f,l_(h))) is a function measuring a loss of a single example; w is a classifier model, which is a vector of feature weights for combinations of values from

and

; w_((f,l) _(r) ₎ is a feature weight for the feature-label pair (f,l_(r)); and C is a regularization constant.
 5. The method of claim 3, wherein outputs derived from the loss function

are generated using an unregularized loss output function l_((f,l) _(u) ₎(u) and a regularization penalty output function r_((f,l) _(u) ₎(u) with respect to potential update values u for the value of a particular feature weight w_((f,l) _(u) ₎.
 6. The method of claim 5, wherein the unregularized loss output 1 is derivable from the formula: ${{l_{({f,l_{u}})}(u)} = {\sum\limits_{{({f,l_{u}^{\prime}})} \in T}{{hl}\left( {{w + {ub}_{({f,l_{u}})}};\left( {f,l_{u}^{\prime}} \right)} \right)}}};\mspace{14mu}{wherein}$ wherein b is a vector having a value of 1 at coordinate (f,l) and a value of 0 at all other coordinates.
 7. The method of claim 5, wherein the regularization penalty output r is derivable from the formula: r _((f,l) _(u) ₎(u)=C|w _((f,l) _(u) ₎ +u|.
 8. The method of claim 2, wherein the loss function

comprises a plurality of losses, each loss of the plurality of losses associated with one of a plurality of training set examples, and at least some of the change in derivative values are computable using individual training set examples.
 9. The method of claim 8, wherein at least some of the plurality of losses are measured using a hinge loss with respect to at least some of the plurality of training set examples, the hinge loss hl(w;(f,l_(h))) measured using a difference between a highest sum of feature weights for a plurality of incorrect labels and a sum of feature weights for a correct label.
 10. The method of claim 9, wherein the hinge loss is determined based on the formula: ${{hl}\left( {w;\left( {f,l_{u}} \right)} \right)} = {{\max\left( {0,{1 + {\max\limits_{l_{h}^{\prime} \neq l_{h}}{\sum\limits_{f \in f}w_{({f,l_{h}^{\prime}})}}} - {\sum\limits_{f \in f}w_{({f,l_{h}})}}}} \right)}.}$
 11. The method of claim 2, wherein the loss function

is not convex and the values are determined for all identified points.
 12. The method of claim 2, wherein the loss function

is convex, the line-searching is performed in order of the identified points along the line, and the line-searching is terminated if the cumulative reduction in loss stops increasing.
 13. The method of claim 2, further comprising: updating a feature weight in a classifier model using the selected point.
 14. The method of claim 2, further comprising: selecting an update set of feature weights from a classifier model to update; determining an update for each of the feature weights included in the update set using steps (A)-(D); determining a step size to modify the magnitude of each of the updates; creating an updated classifier model using the step size and the updates.
 15. The method of claim 14, wherein determining the step size comprises: performing a line search using an unregularized loss output function l(α) and a regularization penalty output function r(α) with respect to values of step size α greater than zero.
 16. The method of claim 15, wherein the output of the loss function l(α) is derivable from the formula: ${{l(\alpha)} = {\sum\limits_{{({f,l_{r}})} \in T}{{hl}\left( {{w + {\alpha\; u}};\left( {f,l_{r}} \right)} \right)}}};{and}$ the output of the regularization function r(a) is derivable from the formula: ${r(\alpha)} = {C{\sum\limits_{f \in}{\sum\limits_{l_{r} \in}{{{w_{({f,l_{r}})} + {\alpha\; u_{({f,l_{r}})}}}}.}}}}$
 17. The method of claim 16, wherein the output of the unregularized loss output function l(α) is determined through the creation of a modified score by subtracting one from each of a plurality of scores associated with a plurality of correct labels in a training set and subtracting the modified score of the correct label from a score of a highest scoring label for the plurality of correct labels for at least some examples in the training set. 